Algebraic hypergeometric transformations of modular origin

Abstract:

It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\leqslant N\leqslant 7$, the $N$-fold cover of $X_0(N)$ by $X_0(N^2)$ gives rise to an algebraic hypergeometric transformation. The $N=2,3,4$ transformations are arithmetic–geometric mean iterations, but the $N=5,6,7$ transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since $X_0(6),X_0(7)$ are of genus $1$. Since their quotients $X_0^+(6),X_0^+(7)$ under the Fricke involution (an Atkin–Lehner involution) are of genus $0$, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.